Thermodynamic Models and Property Estimation

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Thermodynamic Models and Property Estimation

• Pan American Advanced Study Institute
• Workshop on Innovative Approaches to the In-situ
  Assessment and Remediation of Contaminated Sites
• Rio de Janeiro, Brazil
• July 23 August 3, 2002
• Professor Kalliat T Valsaraj
• Gordon A. and Mary Cain Department of Chemical
  Engineering
• Louisiana State University, Baton Rouge, LA 70803

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ENVIRONMENTAL COMPARTMENTSEquilibrium, Reaction
Rates and Mass Exchange.
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EQUILIBRIUM COMPARTMENTS OF CONCERN

• Pure Substance (different phases)
• Air (atmosphere, soil pores, sediment pores)
• Water (ground water, surface water, aerosols)
• Soil/solids (aerosols, soils, sediments,
  adsorbents)
• Biota (organisms, specific organs)
• Vegetation (plants)
• Non-aqueous liquids (oils, lipids)
• Colloids (dissolved organic carbon, particulates)

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Thermodynamic Criteria for Equilibrium Between
Phases

• Chemical potential (?) is an indicator for a
  molecule to move from state a to state b
  identical to hydrostatic potential for fluid
  flow, electrostatic potential for charge flow,
  gravitational potential for mechanical work .
• Chemical potential is the free energy per mole
  available to do chemical work (reactions,
  transport between phases).
• Equality of chemical potential between phases
  means equilibrium between phases.
• Chemical potential is an elusive quantity and is
  only indirectly measured.
• GN Lewis proposed the property fugacity (f) as
  more amenable for equilibrium calculations, since
  it is directly measurable.

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Equilibrium, Chemical Potential and Fugacity
?AG gt ?AL ? G to L transfer feasible
A in the Gas Phase
?AG lt ?AL ? L to G transfer feasible
?AG
?AG ?AL ? Equilibrium
?AL
Fugacity, f ? exp (?/RT)
A in the Liquid Phase
fAG fAL ? Equilibrium
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Fugacity What is it?

• It is derived from the Latin term fugere which
  literally means to flee. As such, it measures
  the escaping tendency of a molecule from a phase.
  It has units of pressure.
• If fugacity in two phases is the same then the
  molecule has reached equilibrium between the
  phases.
• The above thermodynamic criterion allows us to
  obtain the ratios of concentrations at
  equilibrium in a variety of cases.

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Fugacity Definition for Different Phases.

• Fugacity for a gas defined as fi yi ?i P ,where
  yi is the mole fraction of i, ?i is called the
  fugacity coefficient and P is the total pressure.
  For ideal gases ?i 1 and fugacity is the same
  as partial pressure.
• Fugacity for a liquid or solid defined as fi ?i
  xi fi0 where xi is the mole fraction in the
  liquid or solid phase, ?i is the activity
  coefficient (representing the deviation from
  ideality) and fi0 is called the reference
  fugacity. For ideal liquids or solids, ?i 1,
  fi0 P, the pure component vapor pressure and
  fugacity is xiP (Raoults law).

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An Example Gas-Liquid Equilibrium

• Characterized by equal fugacity.
• yi ?iP ?i xi fi0 ? yi / xi ?i fi0 / ?iP .
• The ratio yi / xi is called the equilibrium
  constant.
• Note that mole fractions y and x are proportional
  to concentrations.

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EQUILIBRIUM PARTITION CONSTANT

• Definition

Equilibrium Partition Constant characterizes the
ratio of concentrations of a compound between
any two phases at equilibrium. Keq Ci /
Cj Where Ci and Cj represents the concentration
in the respective phases at equilibrium. The
concentrations may be expressed in various units
appropriate to each phase. Hence the equilibrium
partition constant may have dimensions or be
dimensionless.
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Concentration Units

• Solute in vapor phase partial pressure (pa /
  atm), molar or mass concentration (Ca / mol.m-3
  or mg.m-3), vapor phase mole fraction (y).
• Solute in liquid phase molar or mass
  concentration (Cw / mol.m-3, or mol. L-1 or
  mg.m-3, or mg.L-1 ), weight fraction, mole
  fraction (x).
• Solute in solid phase, colloids or biota mass
  of compound per unit unit mass of solid, colloid
  or biota (W / mg.kg-1), weight fraction.

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Air-Water Equilibrium(Henrys Law)

• (fi)a (fi)w
• (yi?iP) (xi?iPi)
• Note for ideal gas (air) ?i 1 and for compound
  i in water, Pi is the pure component vapor
  pressure. Therefore, yi P xi ?iPi.
• Since yiP Pi represents the partial pressure
  (concentration) in the air and xi represents the
  mole fraction (concentration) in the water, the
  ratio Pi/xi Hi is the equilibrium constant for
  air-water system. This is called the Henrys
  constant.
• Note that Hi is given by ?iPi.
• For most organic compounds that are sparingly
  soluble in water, ?i can be replaced by its value
  at saturation solubility in water, ?i which is
  also given by the reciprocal of the mole fraction
  solubility. In other words, ?i 1/ xi.
  Therefore, Hi Pi / xi.

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Example Benzene in Air-Water Systems

• Saturation solubility in water at 25 C 1,780
  g.m-3
• xi (1,780/78)/(106/18)(1,780/78) 4.1 x
  10-4
• Hence, ?i 1/ xi 2,437.
• Saturation vapor pressure, Pi 0.125 atm.
• Hence, Hi (2,437)(0.125) 304 atm.
• Henrys law constant can also be expressed in
  various other forms.
• Note that to obtain H we need literature data on
  both vapor pressure and saturation aqueous
  solubility.

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Different Forms of Henrys Law

• Definition SI units Relationship to ,i
• ,i Pi/xi Pa --
• ,c Cig/CiR dimensionless ,cvw/ RT,i
• ,x yi/xi dimensionless ,x va/ RT ,i
• ,a Pi/CiR Pa.m3.mol-1 ,a vw/,i
• Note vw is the partial molar volume of water (
  0.018 R/mol at 298K) and va is the partial
  molar volume of air ( 22.4 R/mol at 298K)..
  Cig and CiR are respectively the molar
  concentrations of solute in the gas and liquid
  phases. yi is the mole fraction of solute i in
  the gas phase.

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Estimation of Vapor Pressure

• Antoine equation ln P A B / (t C)
• Compound Range of t A B C
• (EC)
• chloroform -35 to 61 6.493 929.4 196.0
• benzene 8 to 103 6.905 1211.0 220.
• biphenyl 69-271 7.245 1998.7 202.7
• naphthalene 86-250 7.010 1733.7 201.8
• tetrachloroethylene 37-120 6.976 1386.9 217.5
• pyrene 200-395 5.618 1122.0 15.2
• p-dichlorobenzene 95-174 7.020 1590.9 210.2
• pentafluorobenzene 49-94 7.036 1254.0 216.0
• nitrogen 7.345 322.2 269.9
• carbondioxide 9.810 1347.7 273.0
• hydrogen peroxide 7.969 1886.7 220.6
• ammonia 9.963 1617.9 272.5
• sulfur dioxide 7.282 999.9 237.2

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Vapor Pressure Estimation - Continued

• Liquids at room temperature ln Pl ? 19
  1-Tb/T 8.5 ln (Tb/T)
• Solids at room temperature Correction needed
  for sub-cooled liquid. ln (Ps/Ps(l)) ? - 6.8
  Tm/T 1
• Tb is the boiling point of the liquid, Tm is the
  melting point of the solid.

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Vapor Pressure Estimation
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Estimation of Aqueous Solubility

• Remember that for sparingly soluble organic
  compounds, xi 1 / ?i . Hence estimation of
  mole fraction solubility is through activity
  coefficient determination.
• Activity coefficient correlated to the following
• 1. Solute size (molecular area, molecular volume)
• 2. Hydrophobicity indicator (Octanol-water
  partition constant)
• 3. Solute property (normal boiling point)
• 4. Group Contribution methods (UNIFAC)

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Aqueous Solubility Correlation with Molecular Area
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Aqueous Solubility Correlation with Molecular
Volume
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Aqueous Solubility Correlations with
Octanol-Water Partition Coefficients
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Aqueous Solubility from Group Contribution Methods
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Estimation of Octanol.-Water Partition Constants

• Octanol-Water partition constant is a measure of
  the degree of hydrophobicity of a compound. In
  other words, it is a measure of the activity
  coefficient in water.

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Estimation of Octanol-Water Partition Constant
from Group Contributions

• Log Kow ?bj ?Bk
• Tables of data available for fragment constant bj
  and structural factor Bk
• Comprehensive Sources
• 1. C Hansch and A Leo Exploring QSAR, American
  Chemical Society Professional Reference Books,
  ACS, Washington , DC (1995).
• 2. C Hansch, A Leo and D Hoekman Exploring QSAR
  Hydrophobic, Electronic and Steric Constants,
  ACS Professional Books, ACS, Washington, DC
  (1995).

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Estimation of Henrys Constants from Bond and
Group Contribution Methods

• Bond Contribution Scheme Assigns values to each
  bond and sums up the contributions. Reference J
  Hine and P K Mookerjee. Journal of Organic
  Chemistry, Vol. 40, 292- 298 (1975).
• Group Contribution Scheme Breaks up molecule
  into groups, assigns values to each group and
  adds up the contributions. Reference W H Meylan
  and P H Howard. Henrys Law Constant Program,
  Lewis Publishers, Boca Raton, FL (1992).

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Example H from Bond Contribution
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Experimental Values of Henrys Constants
Caution!!
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? A Word of Caution About Physico-chemical Data
Original (a) SW data 9note log scsle), and (b)
log KOW data for DDT (18-25C) plotted as a
function of Publication date (modified from
Pontolillo and Eganhouse, 2001). Double-headed
arrows correspond to data published as a range
of solubilities. Single-headed arrows indicate
data published as maximum solubilities.
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Factors that Effect Experimental Values of Hi

• Temperature
• Co-solvents
• Colloids and Particulates
• pH (for ionic solutes)
• Ionic strength
• Note These effects through their influence on
  aqueous solubility (activity coefficient).

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Soil-Water Partition Constant
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Soils and Sediments have Similar Composition ?
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Soil-Water Partition Constant - Definition

• Ksw (L.kg-1) (W mg solute per kg sorbent) /
  (Cw mg solute per liter of solution)
• Ksw has two contributions (1) mineral surface
  area, (2) Organic carbon fraction on minerals.
• Ksw ?min Sa Kmin ?oc Koc
• In most cases Ksw ?oc Koc, since organic carbon
  dominates partitioning.
• Note that Koc represents the organic carbon
  normalized partition constant and is reasonably
  constant for a chemical over all types of soils
  and sediments.

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Ksw is correlated to Kow (hydrophobic indicator)
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Ksw correlated to Cw
Log Koc a b log Cw
Correlation a b r2 Note
Kenaga and Goring 3.64 -0.55 0.71 Cw in
mg/L Karickhoff et al 0.44 -0.54 0.94 Cw in
mole fraction Chiou et al 4.277 -0.557 0.99 Cw
?moles/L
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Koc Estimation from Correlations Caution!!

• Example benzene (aqueous solubility 1,780
  mg/L).
• Kenaga-Goring correlation 71 mL/g
• Karickhoff et al correlation 186 mL/g
• Chiou et al correlation 71 mL/g
• Variability a factor of 2
• Experimental Value 83 mL/g

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Mineral Matter Contribution to Ksw
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Factors that Effect Ksw

• Colloids in the aqueous phase
  KocKoco/(1CcKc). Note Koco is the soil
  OC-water partition constant without colloids, Cc
  is the colloid concentration, Kc is the
  colloid-water partition constant for contaminant.
  Typically Kc ? Koco. Increased colloid
  concentration decreases partitioning to sediment
  (soil).
• Co-solvents in the aqueous phase ln (Koc/Koco)
  - 0.5 (?? /kBT)(HSA) ?c, Note ?? is the
  difference in surface tension between water and
  pure co-solvent, HSA is the hydrophobic area of
  the solute, ?c is the co-solvent volume fraction.

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Factors that Effect Ksw Continued(Ionizable
Organics, f(pH))
Pentachlorophenol partitioning to soil Lee et al,
1990
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Sources of Koc

• Lyman et al (1990) Handbook of Chemical Property
  Estimation Methods, ACS, Washington, D.C.
• Boethling and Mackay (2000) Handbook of Property
  Estimation Methods for Chemicals Environmental
  and Health Sciences, Lewis Publishers, Boca
  Raton,FL.
• J H Montgomery (1996) Groundwater Chemicals Desk
  Reference, Second Edition, CRC Lewis, Boca Raton,
  FL.
• C L Yaws (1999) Chemical Properties Handbook,
  McGraw Hill, NY.
• M Reinhard and A Drefahl (1999) Handbook for
  Estimating Physicochemical Properties of Organic
  Compounds Toolkit on CDROM, John Wiley, NY.

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Biota-Water Partition Constant
KBW CiB / CiW, where CiB is the mass of
pollutant per mass of the organism.
Most of the pollutant is associated with the
lipid mass of the organism.
Since octanol is a good surrogate for the lipid
content, KBW is correlated with the
octanol-water partition constant. Log KBW a
log Kow b.
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Bioconcentration Factor Another name for
Biota-water Partitioning

• (fw) (fj) where j represents any one
  compartment within the organism.
• Thus, for biota-water Cwvw?wfo Cjvj?jfo
• Note that CB ? Cj
• Similarly for the octanol-water system Cwvw?wfo
  Covo?ofo.
• Leads to KBW Constant . Kow

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Correlation to Estimate KBW
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Soil (Sediment) Air Partition Constant
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Soil (Sediment) - Air Partition Constant

• Dry soil - lt 2 w/w water content - Soil
  sorption controlled by air-soil partitioning -
  BET multilayer adsorption S shaped isotherm.
• Damp soil - 2 5 water content - Soil
  sorption from mixed phases (air-soil-water).
• Wet soil - gt 5 water content Soil sorption
  controlled by water-soil partitioning
  Vaporization controlled by water vapor
  partitioning - Soil to Water to Air pathway.
• Contaminant partial pressure in soil air Wet
  soil gtgt Damp soilgtgt Dry soil.

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Equations for Estimating Soil (Sediment)-Air
Partition Constants

• Dry soil - KSA (KL/ Ca) Wimax
• Damp soil KSA (KL/ Ca) Wimax 1 - ?w
• Wet Soil - KSA Ksw / Kaw
• KL Langmuir adsorption constant for
  contaminant.
• Ca - Saturation vapor concentration of
  contaminant.
• Wimax - monolayer adsorption capacity for
  contaminant on soil
• ?w - fractional soil surface coverage by water.
• KSW soil-water partition constant for
  contaminant.
• KAW dimensionless molar concentration ratio
  Henrys constant for contaminant between air and
  water.

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Relative Magnitudes of Sediment-Air Partition
Constants for Different Moisture Sediments.
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Air-to-Vegetation Partition Constant

• Plant surfaces have wax or lipid layers to
  prevent excessive evapo-transpiration. They also
  accumulate organic compounds.
• Partition constant defined as KVA Wv/(?LCa)
• Wv is concentration in vegetation (ng/g dry
  weight), ?L is the lipid content (mg/g dry
  weight) and Ca is the atmospheric concentration
  of contaminant (ng/m3).

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Correlations for KVA as f(T)
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Non-Aqueous Liquids and Equilibrium

• NALs prevalent in the environment in groundwaters
  and sediments.
• Both NAL air and NAL water interfaces are
  important to recognize.
• Can use the principle of equal fugacity at
  equilibrium to derive the concentrations in the
  adjacent medium.

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NAL Water Equilibrium

• Example gasoline-water system.
• fi xiw?iwfil,o xio?iofil,o
• If NAL is ideal ?io is 1.
• For pure i, saturation solubility in water is xi
  1/ ?i. If interaction in water is small ?iw ?
  ?i.
• Hence, xiw / xi xio

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NAL Equilibrium Example Benzene from gasoline
into water

• Gasoline benzene mole fraction xio , activity
  coefficient, ?io ? 1 (ideal solution) and 1.5 (
  theoretical derived from the Scatchard
  Hildebrand theory considering octane properties
  for gasoline).
• Water mole fraction of benzene xiw, activity
  coefficient, ?io ? 2,400 based ib saturation
  solubility in water.
• For ideal solution, ?io 1. Hence, xio
  (2,400/1 ) (xiw).
• Assume gasoline contains 1 benzene (vol ? wt ,
  average mole. Wt of gasoline 100)
• xio (0.01/78)/(0.99/100)(0.01/78) 0.013
• xiw 0.013/2,400 5.3 x 10-6
• Ciw xiw (106/18) 78 23 g/m3
• Note that for non-ideal solution with ?io
  1.5, Ciw 35 g/m3.

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NAL Spill scenario
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NAL Air Equilibrium

• Example gasoline-air system.
• fi yiP xio?iofil,o . Note fil,o P.
  Reference fugacity for gas phase is P 1 atm.
• If NAL is ideal ?io is 1.
• Hence, yi xio (P/P). This is Raoults law.
• Note that the above is valid only if NAL mixture
  is composed of similar organics. E.g. hexane in
  mixture of n-alkanes. BTEX in gasoline is
  marginally ideal.

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Colloids in Natural Water
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Colloid -Water Equilibrium
Diffusion from Sediment Bed to Overlying Water.
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Colloid - Water Equilibrium
Volatilization from a Contaminated Water Body.
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Colloid -Water Equilibrium

• KC represents the equilibrium concentration ratio
  of contaminant between colloid and water.
• In most cases, since colloids in water are mostly
  high molecular weight dissolved macromolecular
  species (humic and fulvic acids), KC KOC.
• Transport into overlying water is facilitated by
  colloids in a sediment-water system. Deff Ds
  DCKCCC.
• Volatilization from a water body is reduced since
  effective water concentration is reduced by
  colloids. Cw,eff Cw /1 KCCC.

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Colloid Water Partition Constants
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Applications of Equilibrium Partition Constants
for Environmental Calculations

• Mackay (1979) first suggested the use of fugacity
  based models to estimate the partitioning of
  chemicals between environmental compartment.
• To relate fugacity (Pa) to concentration (mol.
  m-3) in each compartment, a fugacity capacity, Z
  (mol.m-3. Pa-1) is defined
• Z C / f
• A table of values for Z for various compartments
  were generated in which each of the respective
  equilibrium partition constants were provided.

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Fugacity Capacity Definition
60
Fugacity Level I Model

• Once the fugacity capacities are known for
  individual compartments, the mean Z value can be
  determined by multiplying the respective Zj value
  with the volume of the compartment (vj) and
  summing over all compartments, ie., ?vjZj
• If the total mass of the compound (M, mol) or
  chemical input or inventory in all the
  compartments is known, M ?mj, then the
  fugacity of the compound is given by
• f M / ?vjZj
• The respective individual concentration in each
  compartment is then given by
• Cj f Zj

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Fugacity Level I - Example

• Consider an evaluative environment consisting of
  air, water, soil and sediment.
• The volumes of the phases are Air 6x109 m3,
  Water 7x106 m3, Soil 4.5x104 m3 and Sediment
  2.1x104 m3.
• The properties for pyrene are Henry's constant
  0.9 Pa.m3.mol-1 Kd (soil) 1.23x103 R.kg-1 Ds
  (for sediment and soil) 1.5x10-2 kg.R-1 Kd
  (sed) 2.05x103 R.kg-1.
• Let the temperature be 300K and the total
  inventory of pyrene be 1000 mol.
• Determine the equilibrium distribution of a
  hydrophobic pollutant such as pyrene in this four
  compartment model.

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Fugacity Level I Example Continued

• Step 1 Calculate the fugacity capacity Z as
  follows
• Air Z1 1/RT 1/(8.314 x 300) 4 x 10-4
• Water Z2 1/H 1/0.89 1.1
• Soil Z3 KdDs/H (1.23x103)(1.5x10-2)/0.89
  20.7
• Sediment Z4 (2.05x103)(1.5x10-2)/0.89 34.5
• Step 2 Calculate the fugacity f
• f M / ? vj Zj 1000/1.2x107 8.5x10-5 Pa

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Fugacity Level I Model - Continued

• Step 3 Calculate the concentrations in each
  phase, Cj
• Cair fZ1 3.4 x 10-8 mol.m-3
• Cwater fZ2 9.4 x 10-5 mol.m-3
• Csoil fZ3 1.7 x 10-3 mol.m-3
• Csediment fZ4 2.9 x 10-3 mol.m-3
• Step 4 Calculate the total mass of pyrene in
  each compartment,mj
• mair Cairvair 205 mol
• mwater Cwatervwater 658 mol
• msoil Csoilvsoil 76 mol
• msediment Csedimentvsediment 61 mol
• Conclusion The largest fraction (65.8) of
  pyrene is in water. The next largest fraction
  (20.5) resides in the air. Both sediment and
  soil environments contain less than 10 of pyrene
  each.

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Fugacity Level II and III Models

• Level II is the next higher level in fugacity
  models. It considers inflows and outflows from
  each compartment, but assumes equilibrium
  distribution of chemicals across all
  compartments.
• Level III relaxes the assumption of equilibrium
  distribution in all compartments and considers
  mass transfer rates between compartments.
• Level III model is widely used in a number of
  equilibrium partitioning calculations in the
  environmental context.
• All models are available in user friendly
  versions at the web site http//www.trentu.ca/cem
  c.
• The book by D Mackay Multimedia Environmental
  Models The Fugacity Approach, Second Edition,
  Lewis Publishers, Boca Raton, FL (2001) provides
  more details.

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Comprehensive Sources of Physicochemical Property
Data
66
Books on Property Estimations

• M Reinhard and A Drefahl Handbook for Estimating
  Physicochemical Properties of Organic Compounds
  Toolkit on CDROM, John Wiley, NY (1999).
• B E Poling, J M Prausnitz and J P OConnell The
  Properties of Gases and Liquids, Fifth Edition,
  McGraw Hill, NY (2000).
• K T Valsaraj Elements of Environmental
  Engineering Thermodynamics and Kinetics, Second
  Edition, CRC/Lewis Publishers, Boca Raton, FL
  (2000).
• P M Boethling and D Mackay Handbook of Property
  Estimation Methods for Chemicals, Lewis
  Pubishers, Boca Raton, FL (2001).
• W J Lyman, W F Reehl and R H Rosenblatt Handbook
  of Chemical Property Estimation Methods, ACS,
  Washington, DC (1990).
• R P Schwarzenbach, P M Gschwend and D M Imboden
  Environmental Organic Chemistry, John Wiley, NY
  (1993).
• R Baum Chemical Property Estimation Theory and
  Applications, Lewis Publishers, Boca Raton, FL
  (1998).

67
Web Sites for Environmental Property Data

• http//www.chemfinder.camsoft.com
• http//www.webbook.nist.gov
• http//www.syrres.com
• http//www.epa.gov
• http//www.dep.state.pa.us/physicalproperties
• http//www.cas.org/online/dbss/dipprss.html